Kubrak–Travkin's conical resolution inequality

Let $\mathcal O_K$ be the ring of integers of a finite extension of $\mathbb Q_p$, let $k$ be its residue field, and let $\pi\colon X\to Y$ be a conical resolution over $\mathcal O_K$. The notation $X(\mathbb C)/\mathbb C^\times$ denotes the homotopy quotient. Kubrak–Travkin's conjecture. For every $i\geq 0$,

$$\dim_k H^i_{\mathrm{dR}}([X/\mathbb G_m]_k/k)\geq \dim_{\mathbb F_p}H^i_{\mathrm{sing}}(X(\mathbb C)/\mathbb C^\times,\mathbb F_p).$$

The $i=1$ case was attributed to Roman Travkin and the first author as a generalized version of an earlier conjecture. The displayed inequality is the proposed extension to all degrees and remains open in the supplied text.