Conjecture on projective dimension of the exceptional line in an elliptic blowup

Let $T$ be an elliptic algebra with $\deg T \geq 4$, let $p \in E$, and let $T(p)$ be the blowup of $T$ at $p$, with exceptional line $L$. Suppose there is no $T$-line module $L'$ satisfying

$$\operatorname{Div} L' = \tau(p).$$

Exceptional-line projective-dimension conjecture. Then

$$\operatorname{pdim}_{T(p)^\circ}L^\circ=1.$$

The conjecture would characterize, in the stated situation, when the exceptional line has finite projective dimension; the source gives no resolution, so its status remains open.