Conjectured Hilbert-Schmidt radius for absolutely PPT qutrit-qudit states

Let $M_3\otimes M_n$ be equipped with the Hilbert-Schmidt norm, and let $I/(3n)$ denote the maximally mixed state. Consider the smallest Euclidean ball centered at $I/(3n)$ that contains all absolutely PPT states in a $3\otimes n$ system. Geometric-radius conjecture. Its radius is

$$\begin{cases} \frac1{3\sqrt n} &\text{if } n\equiv 0\pmod 4,\\ \frac{\sqrt{27n^2+6n-1}}{(9n+1)\sqrt{3n}} &\text{if } n\equiv 1\pmod 4,\\ \frac{\sqrt{27n^2+12n-4}}{(9n+2)\sqrt{3n}} &\text{if } n\equiv 2\pmod 4,\\ \frac{\sqrt{27n^2-6n-1}}{(9n-1)\sqrt{3n}} &\text{if } n\equiv 3\pmod 4. \end{cases}$$

This is the geometric reformulation of the conjectured maximum-purity formula for absolutely PPT qutrit-qudit states; the numerical estimates do not establish the formula, so it remains open.