Open problems of Bruckamp, Caicedo, and Juhnke on Hermite normal form simplices

For the Hermite normal form simplices $S_{\boldsymbol a}=\operatorname{conv}(0,e_1,\ldots,e_{d-1},\boldsymbol a)$, where $\boldsymbol a=(a_1,\ldots,a_{d-1},N)$, and $T_{d,N}=\operatorname{conv}\bigl(0,e_1,\ldots,e_{d-2},(d-2,\ldots,d-2,d-1,0),(1,\ldots,1,N)\bigr)$, determine: (1) whether the Ehrhart-coefficient sequence of $T_{d,N}$ is unimodal for all admissible $d$ and $N$, and classify the cases in which the relevant Ehrhart data are log-concave or real-rooted; (2) for the subfamily $S_{\boldsymbol a}$ with $\boldsymbol a=(N-q,\ldots,N-q,N)$, characterize exactly when $S_{\boldsymbol a}$ has the integer decomposition property and when it admits a regular unimodular triangulation, in terms of a congruence condition on a negative continued fraction; and (3) for arbitrary admissible $\boldsymbol a$, characterize when $S_{\boldsymbol a}$ has the integer decomposition property and when it admits a unimodular triangulation.