We will probably never know. No evidence.
Historians will normally stress that what's important is not whether claims about (im-)morality of rulers are factually true, but whether people actually believe that these claims are true. Popular rumours about the Romanov family during 1WW played a substantial part in the overthrow of the regime in 1917.
Orlando Figes, A Russian Revolution Scholar writes in his 2014 book Revolutionary Russia, 1891-1991:
Similar credence was given to the rumours of sexual scandals at the
court. It was said that the Empress was the mistress of Rasputin and
the lesbian lover of Anna Vyrubova, her lady-in-waiting, who took part
in orgies with them both. Alexandra’s ‘sexual corruption’ became a
kind of metaphor for the diseased condition of the monarchy. Similar
pornographic tales about Marie Antoinette and the ‘impotent Louis’ had
circulated on the eve of the French Revolution in 1789. None of these
rumours had any basis in fact (Vyrubova was a dim-witted spinster
infatuated with the mystical powers of Rasputin and medically
certified to be a virgin by a special commission appointed to
investigate the charges against her in 1917). But the point of the
rumours was not their truth or untruth: it was their power to mobilize
an angry public against the monarchy. In a revolutionary crisis it is
perceptions and beliefs that really count. Without this ‘atmosphere’ –
created out of gossip, half-truths, facts and fabrications, bits of
information from the press which were then distorted into fantasies –
it is impossible to understand the ‘revolutionary mood’ or the ways in
which the revolution turned on the interpretation of hearsay and
events.
Source: Revolutionary Russia, 1891-1991: A Pelican Introduction
