Bjerkevik's stability conjecture for the pruning distance

Let $M$ and $N$ be pfd modules, and let $d_P$ denote the pruning distance and $d_I$ the interleaving distance. For a module $M$, write

$$\operatorname{supdim} M=\sup_{p\in\mathbb{P}}\dim M_p.$$

Bjerkevik's conjecture. On persistence modules of bounded pointwise dimension, the pruning distance is stable and Lipschitz equivalent to the interleaving distance. More precisely, if $r=\operatorname{supdim}M<\infty$, then

$$d_P(M,N)\leq d_I(M,N)\leq 2r\,d_P(M,N).$$

The left-hand inequality is known when the pruning parameter is restricted to $\delta=0$, but its extension to arbitrary positive parameters remains open; the stated stability and Lipschitz-equivalence conjecture is therefore open.