Strict separation between regularized Rains and PPT relative entropy

Let $v_j=(|0,j\rangle-|j,0\rangle)/\sqrt2$ for $j=1,2$, using coordinate basis kets on $\{0,1,2\}\times\{0,1,2\}$, and let $\rho_V=\tfrac12(v_1v_1^\dagger+v_2v_2^\dagger)$, with outer-product entries $v_x\overline{v_y}$. The assertion is the extended-real conjunction

Here $\Gamma$ transposes the second party, $(\Gamma X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. Define $D_2(X\|S)=\operatorname{Re}\operatorname{Tr}(X\log X-X\log S)/\log2$ when $\ker S\subseteq\ker X$, and $+\infty$ otherwise, using spectral functional calculus and $\log0=0$. Then

A channel is a finite Kraus sum $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$, and it is PPT if its input-output partial-transpose conjugate is also a channel. The quantity $D_{\mathrm{PPT}}(\rho_V)$ is the supremum of real $r\ge0$ such that, for every $\varepsilon>0$, all sufficiently large natural $n$ admit a PPT channel mapping $\rho_V^{\otimes n}$ within trace norm $\varepsilon$ of $\Phi_{\lceil2^{nr}\rceil}$. The quantity $C_{\mathrm{PPT}}(\rho_V)$ is the infimum of real $r\ge0$ such that, for every $\varepsilon,\delta>0$, all sufficiently large positive natural $n$ admit $m\in\mathbb N$ and a PPT channel mapping $\Phi_{2^m}$ within trace norm $\varepsilon$ of $\rho_V^{\otimes n}$, with $m\le n(r+\delta)$. The matrix $\Phi_k$ has entry $1/k$ when both row and column have equal party indices and $0$ otherwise; tensor-power entries are products, party registers are grouped and relabelled by fixed finite-index bijections, and $m=0$ is included.