Unboundedness of polynomial ergodic averages' variation operator on $L^p$

Let $(X,\mathcal{B},\mu,T)$ be the measure-preserving system under consideration, let $P(n)$ be an integer-valued polynomial, and for $N\geq 1$ write

$$M_Nf(x)=\frac{1}{N}\sum_{n=1}^N f\bigl(T^{P(n)}x\bigr).$$

For a sequence $(a_N)$, let $\mathcal{V}^2(a_N)$ denote its 2-variation, and apply this pointwise to $(M_Nf(x))_{N\geq 1}$. Unboundedness conjecture. For each $1\leq p\leq\infty$, and any integer-valued polynomial $P(n)$, the operator

$$\mathcal{V}^2(M_Nf)$$

is unbounded on $L^p(X)$. This question concerns whether the variation estimate known for the relevant $L^p$ setting can extend beyond the established range; the source presents the assertion as a conjectural risk motivated by the poor behavior of the analogous operator for standard Birkhoff averages, and does not give a resolution.