Relatives of the first Russian soldier to be confirmed dead in Syria buried the 19-year-old man, as more doubts emerged about the exact cause of his death.
The Chief Military Prosecutor's Office announced on October 28 that its preliminary findings agreed with the Defense Ministry's statement that Kostenko had hanged himself because of problems in a personal relationship.
It also said the probe was continuing and "all the circumstances of the serviceman's death will be investigated."
Kostenko's death comes as the Russian air campaign in support of Syrian President Bashar al-Assad nears its fourth week.
The issue of Russian soldiers' deaths has been problematic in the past for the Kremlin, particularly in eastern Ukraine where Moscow has strenuously denied that its troops have fought alongside separatists despite what Kyiv and NATO say is incontrovertible evidence to the contrary.
With Russian soldiers being buried in Russian graves and relatives demanding full acknowledgment of their sons' participation in combat, President Vladimir Putin signed a law earlier this year that expanded prohibitions on reporting about military deaths.
The deployment in Syria, by contrast, is Russia's largest outside of the former Soviet Union in decades, and has been trumpeted by state-controlled media as indicative of Russia's returning might.
The Defense Ministry has given foreign media tours of the Hmeimim air base, in the Syrian western province of Latakia, showcasing the conditions that Russian troops are working under.
Also on October 28, friends and relatives mourned Kostenko in a ceremony attended by hundreds in the southern Russian region of Krasnodar.
Sobbing mourners accompanied a coffin as it was carried through the streets by what appeared to be Russian soldiers. A woman with the Russian tricolor flag also accompanied the procession.
Kostenko's relatives have openly doubted the official conclusion that he killed himself, and requested an independent autopsy, saying they spoke with him by telephone the day of his death and that he was cheerful.
There were "no marks on his body to suggest that he had hanged himself," Kostenko's father, Aleksandr, said.
WATCH: Russian Soldier's Friends React To Suicide Claims
A Reuters reporter who attended the burial at the village cemetery, said the man's coffin was opened briefly. His collar was turned up, which concealed his neck, but a bruise was visible on his nose.
The Russian investigative newspaper Novaya Gazeta, meanwhile, quoted Kostenko's uncle as saying the body showed injuries to the skull and neck, with a broken nose and jaw.
Social media posts compiled by Russian bloggers indicated that Kostenko served as part of the 960th Aviation Attack Regiment of the 1st Guards Mixed Air Division, based at the Primorsko-Akhtarsk air base, also in the Krasnodar region.
They said Kostenko had earlier this year completed his mandatory conscription, which is required of all able-bodied Russian men aged between 18 and 27, and had then signed a contract to serve further as a "kontraktnik" -- a volunteer or private soldier.
After weeks of a stealth build-up, which included heavy artillery, advanced tanks, fighter jets, and military personnel, Russian forces launched air strikes on September 30, using jets and later, cruise missiles fired from ships in the Caspian Sea.
The Kremlin has said its campaign is targeting terrorists. However, the United States and others, who are waging their own air campaign against Islamic State (IS) militants, say the Russian strikes are hitting mainly non-IS targets, including moderate rebels, some of whom have been supplied with U.S. weaponry.
U.S. officials have said Moscow's real aim is to prop up the government of Syrian President Bashar al-Assad, a longtime ally.
Last week, Reuters cited Syrian activists as saying three Russians had died when a mortar shell hit their position in Latakia. A Kremlin spokesman later denied that report.
With reporting by Reuters