The support lower bound can fall below PPT distillable entanglement

Let $\rho_-$ be the matrix on $\mathbb C^3\otimes\mathbb C^3$ whose entries are

Write $(\Gamma X)_{(a,b),(c,d)}=X_{(a,d),(c,b)}$ and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. Let $P_\rho$ be the functional-calculus value at $\rho$ of the scalar function that is zero at zero and one elsewhere, and define $E_\eta(\rho)=-\log_2\eta(\rho)$, where $\eta(\rho)$ is the infimum of $\|\Gamma Y\|_{\mathrm{op}}$ over Hermitian $Y$ satisfying $Y-\Gamma P_\rho\succeq0$ and $Y+\Gamma P_\rho\succeq0$. A PPT channel means a map $\Phi$ for which both $\Phi$ and $\Gamma\circ\Phi\circ\Gamma$ have finite Kraus representations with the sum of $K_j^\dagger K_j$ equal to the identity. Define $D_{\mathrm{PPT}}(\rho)$ as the extended-real supremum of nonnegative real $r$ such that, for every $\tau>0$, all sufficiently large natural $n$ admit a PPT channel taking $\rho^{\otimes n}$ within trace norm $\tau$ of $\Omega_{\lceil2^{nr}\rceil}$; define $C_{\mathrm{PPT}}(\rho)$ as the extended-real infimum of nonnegative real $r$ such that, for every $\varepsilon,\delta>0$, all sufficiently large positive $n$ admit a natural $m\le n(r+\delta)$ and a PPT channel taking $\Omega_{2^m}$ within trace norm $\varepsilon$ of $\rho^{\otimes n}$. Here tensor-power basis indices are relabelled by the specified finite-set bijections, and $\Omega_k$ has entry $1/k$ when both its row and column pairs have equal components, and zero otherwise; $m=0$ and rate zero are allowed. The assertion is

with real values embedded into the extended real numbers where required.