The v-number conjecture for monomial ideals with linear powers

Let $S$ be a standard graded polynomial ring over a field, and let $I\subset S$ be a monomial ideal. Write $\alpha(I)$ for the initial degree of $I$, and let $\v(I^k)$ denote the v-number of the $k$th power of $I$. Say that $I$ has linear powers if every power $I^k$ has a linear resolution.

The v-number conjecture. If $I$ has linear powers, then

$$\v(I^k)=\alpha(I)k-1\qquad\text{for all }k\ge1.$$

For ideals with linear powers, this predicts that the v-number always attains its minimal possible value. The statement is false for general graded ideals with linear powers, as shown in the source by $(x^2+y^2)\subset\mathbb{Q}[x,y]$, but the monomial case remains unresolved in the supplied text.