No, not exactly. In reality the software worked as intended and sampled an unidentified program that it considered malicious. It just so happened to be a piece of NSA malware contained on a NSA employee's computer (who re-enabled KAV on his machine after infecting it with a fake Office activator). The US government needed a quick scapegoat and thus they picked the spooky russian company instead of their own employee.
After the allegations were made, Kaspersky opened itself up for third-party auditing of its internal processes, relationship with the government and the events surrounding the above, of which it provides regular updates of: https://www.kaspersky.com/blog/internal-investigation-prelim... https://www.kaspersky.com/blog/transparency-status-updates/2...
They're also moving all data processing to Switzerland, way outside of the reach of the Russian government.