The poll question proved distasteful to many Russians, for whom the nearly 900-day Leningrad siege -- in which an estimated 1 million people died of starvation -- remains one of the grimmest chapters of the country's World War II history.
Dozhd quickly removed the question and apologized for causing offense. But it was not enough to prevent a number of broadcasters in Russia -- including the Akado and NTV-Plus cable providers -- from taking Dozhd off the air.
Later, Valery Kostarev, a representative of the state telecoms operator Rostelekom, announced on January 30 that it would suspend further broadcasts until the "extremely negative situation" with Dozhd was resolved. President Vladimir Putin's spokesman, Dmitry Peskov, said the station had "crossed every line of acceptable behavior."
Pavel Lobkov, a veteran television host who moved to Dozhd after being fired in 2012 by the pro-government NTV station for what he described as "excessive political activity," told RFE/RL correspondent Vladimir Kara-Murza that the poll scandal was merely a pretext for bumping Dozhd off the air.
"The fact that a question about the blockade suddenly became so important -- these are normal political methods," he said. "When you want to get the widest possible audience to support you in shutting down a channel, you don't want to tap into their minds, but into their subconscious. And these so-called taboo themes are the way to do that with most people. Taboo themes are the blockade, some aspects of the [Bolshevik] Revolution, and the mass repressions of the 1930s, which we now refer to as the too-high price for the effective modernization of the country."
Until the poll controversy, Dozhd was one of the few remaining broadcasters still operating outside the highly managed arena of Russia's state-controlled media.
Since its launch in 2010, Dozhd has attempted to bridge the divide between critical reporting and popular culture, providing live feeds from public protests while capitalizing on the celebrity appeal of talk show hosts like gadfly Kseniya Sobchak.
For Russia's young, politically minded elite, as well as Russia-watchers outside the country, Dozhd was considered essential viewing during the antigovernment protests in 2011 and 2012 that rocked the country in the run-up to Putin's controversial reelection.
It also gave airtime to prominent opposition figures like anticorruption crusader Aleksei Navalny, who are shunned by Kremlin-controlled broadcasters.
The reported closures don't necessarily spell the death knell for Dozhd, which enjoys a lively online presence and continues broadcasts through a number of operators.
But some challenges await: In addition to being taken off the air by some cable providers, Dozhd now faces a potential investigation by Russian Prosecutor-General Yury Chaika at the behest of St. Petersburg lawmakers angry over the poll.
Political analyst Boris Vishnevsky, a St. Petersburg native whose parents lived through the blockade, says he personally found the question offensive, but adds that Dozhd broke no laws by conducting the poll.
"The lawmakers who appealed to Prosecutor-General Chaika couldn't name a single norm or law that Dozhd allegedly violated by publishing that poll," he says. "There is no law that prohibits asking questions of one kind or another. Also, I'll remind my colleagues that this question [about the blockade] isn't new. Thirty years ago, the great Russian writer Viktor Astafyev wrote an entire book based on the notion that Leningrad should have been surrendered. Are we going to have prosecutors write to Astafyev posthumously and ask to remove his books from the library, are we going to posthumously hold him to account? It's obvious that this is totally absurd!"
[UPDATE: Lenta.ru reported on January 31 that prosecutors said they had found no extremism in the Leningrad program.]
The Kremlin has repeatedly put the squeeze on opposition media in an effort to manicure its public image. In 2013, Russia ranked 148th out of 179 countries in the press-freedom index released by Reporters Without Borders.
The apparent Dozhd crackdown comes amid two major news events that the Kremlin is seen as particularly keen to spin its own way: the upcoming Sochi Winter Olympics, and the roiling political protests in neighboring Ukraine.
Philosopher Igor Chubais says the sight of organized, effective antigovernment protests -- motivated in part by the Kremlin's own meddling -- is something Putin is desperate to keep off television screens in Russia.
By removing Dozhd, which had provided steady Euromaidan coverage, Chubais says Putin can better control the message on Ukraine -- and prevent a similar revolutionary fever from breaking out in his own country.
"The Ukraine situation is heating up," he says. "Yesterday [January 28], Putin spoke in Brussels and he said that the worst, most dangerous thing would be to resolve the issue by military force. If things are really moving in that direction, then he may feel that the only channel that gives fairly complete, albeit sometimes contradictory, information should be taken off the air."