This Skulk's Special Trait lists 3 conditions that can see through their invisibility (mirrors, special candles and preteens), but it doesn't list Truesight as one of the means, and further is called Fallible Invisibility.
Yes, it would.
The description of the skulk's Fallible Invisibility trait says (Mordenkainen's Tome of Foes, p. 227):
The skulk is invisible. This invisibility can be circumvented by three things:
- The skulk appears as a drab, smooth-skinned humanoid if its reflection can be seen in a mirror or on another surface.
- The skulk appears as a dim, translucent form in the light of a candle made of fat rendered from a corpse whose identity is unknown.
- Humanoid children, aged 10 and under, can see through this invisibility.
The Skulk's invisibility can be circumvented by those means, which means that the invisibility fails to be effective in such conditions. However, characters with truesight can see invisible targets, so they do not need to circumvent it, as they could already see it normally.
The description of Truesight says (emphasis mine):
A monster with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them, and perceive the original form of a shapechanger or a creature that is transformed by magic. Furthermore, the monster can see into the Ethereal Plane within the same range.