Nesbitt–Shapiro inequality at the remaining endpoint values

Let $n$ be a positive integer, let $p$ be the number of positive definite matrices in the cyclic sum, and let $F_p$ denote the corresponding cyclic expression.

Nesbitt–Shapiro endpoint conjecture. If $p=12$ or $p=23$, then

$$F_p\geq \frac{p}{2}n.$$

The paper explains that this would, together with the preceding proposition and the known cases, establish the inequality for all remaining values in the stated ranges. The parser provides no evidence that these two cases have been resolved, so their status is open.