Buck et al.'s three-run enumeration conjecture for flattened 2-Stirling permutations

Let $\mathcal{Q}_n$ denote the set of $2$-Stirling permutations of order $n$, and let $\operatorname{flat}_3(\mathcal{Q}_n)$ denote those with exactly three runs. Then $|\operatorname{flat}_3(\mathcal{Q}_n)|$ is given by

$$\begin{aligned} |\operatorname{flat}_3(\mathcal{Q}_n)|={}&2\sum_{i=1}^{n-1}\binom{n-1}{i}\left(\sum_{j=2}^{n-1-i}\binom{n-1-i}{j}\right)\\ &+\sum_{i=2}^{n-1}\binom{n-1}{i}\left(\sum_{j=2}^{n-1-i}\binom{n-1-i}{j}\right)+\sum_{i=3}^{n-1}(2^{i-1}-2)\binom{n-1}{i}. \end{aligned}$$

Buck et al.'s three-run enumeration conjecture. The number of flattened $2$-Stirling permutations of order $n$ with exactly three runs equals the displayed sum.

The formula was conjectured in the cited work and is presented here as a conjecture; the source gives no further resolution status.