The tightness conjecture for the learning coefficient in factor analysis

Let $p$ and $k$ be positive integers with $p>k$, and let $r\in\{0,\ldots,k\}$ satisfy the assumptions of Theorem bound-alternate, namely, for the learning coefficient $\ell_k(\Sigma_0)$ at a fixed generic covariance matrix $\Sigma_0$ in the $r$-factor model,

$$d_r=p(r+1)-r(r-1)/2\leq p(p+1)/2.$$

Tightness conjecture. The bound from Theorem bound-alternate is tight for all such $p$, $k$, and $r$; equivalently,

$$\ell_{kr}=\frac{p(k+2)+r(p-k+1)}{4}.$$

The conjecture asserts equality in the preceding upper bound throughout the stated range of factor-analysis parameters. The parser provides no resolution evidence, so its status remains open.