Reciprocal-integer deficiency conjecture for extremal matrices

Let $A$ be an extremal matrix, and let $\delta$ denote its deficiency.

Reciprocal-integer deficiency conjecture. There exists an $m\in\mathbb{N}$ such that

$$\delta=\frac{1}{m}.$$

This conjecture predicts that deficiencies of extremal matrices are always reciprocals of positive integers. The supplied source gives no resolution status.