Endpoint boundedness conjecture for Shimorin-type operators

Let $\nu$ be a finite positive Borel measure on $[0,1]$, let $T_\nu:L^p(\mathbb{D})\to L^q(\mathbb{D})$ be the associated Shimorin-type operator, and let $\mathcal{C}$ be the boundary line

$$\mathcal{C}=\left\{\left(\frac1p,\frac1q\right):1\leq p\leq c_\nu',\quad \frac1q=\frac1p+\frac1{c_\nu}-1\right\},$$

with $c_\nu$ defined by the critical integrability index in the supplied context. Endpoint boundedness conjecture. The operator $T_\nu$ is bounded on the interior of the boundary line $\mathcal{C}$, while it is not bounded at the endpoints of the boundary line. This conjecture refines the proposed $L^p$-$L^q$ phase diagram by distinguishing interior points from endpoints; the supplied text gives no resolution, so the endpoint behavior remains open.