Conjecture on the explicit slope-and-$\mathscr L$-invariant list formula

Let $p$ be a prime and $N=1$. For a $p$-new eigenform $f\in S_k(\Gamma)$ of slope $h$, let $\mathscr L_f$ be its $\mathscr L$-invariant and define

$$\widetilde m_p(f)=\max\left(\left\lfloor\widetilde m_p(h)\right\rfloor,\left\lceil-v_p(\mathscr L_f)\right\rceil\right).$$

Here $\widetilde m_p(h)$ is a non-negative function. Explicit slope-and-$\mathscr L$-invariant list conjecture. For $3\leq p\leq 11$, Question 4.9 has an affirmative answer witnessed by

$$\widetilde m_p(h)=\left\lfloor\log_p(h)\right\rfloor+1$$

when $h\geq1$; equivalently, the list of these $\widetilde m_p(f)$ values equals the list of integers $j$ for which $d_j(k)>d_{j-1}(k)$, with multiplicity $d_j(k)-d_{j-1}(k)$. The source presents this as a falsifiable conjecture supported by numerical data, while the immediately preceding question asks whether some such function exists; no proof is given.