Conjecture on the smallest nontrivial defect degree and block height

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$-block of $G$ with defect group $D$. Write $\operatorname{ht}(B)$ for the set of heights of the irreducible characters in $B$. Smallest-nontrivial-element conjecture. Then

$$\inf(\operatorname{cd}(D)\setminus\{1\})=p^{\inf(\operatorname{ht}(B)\setminus\{0\})}.$$

The source presents this as an additional proposed relation between defect-group character degrees and block heights; no general proof or disproof is given.