Kuhl et al.'s existence conjecture for two-repetition gerechte designs with five subsquares

Let $h_1,\dots,h_5$ be positive integers, let $n=h_1+\cdots+h_5$, and let $[5]=\{1,2,3,4,5\}$. A $\operatorname{2-RP}(h_1\dots h_5)$ is a two-repetition gerechte design whose five subsquares have side lengths $h_1,\dots,h_5$. Kuhl et al.'s existence conjecture. A $\operatorname{2-RP}(h_1\dots h_5)$ exists if and only if

$$n^2 - \sum_{i=1}^5h_i^2\geq 3\left( \sum_{i\in D}h_i\right)\left(\sum_{j\in\overline{D}}h_j\right)$$

for all subsets $D$ of $[5]$ where $|D| = 3$. This gives necessary and sufficient conditions for the existence of such designs with five subsquares; the source attributes the conjecture to Kuhl et al., and no resolution is supplied here.