Characters for Projective Modules in the BGG Category $\mathcal{O}$ for the Orthosymplectic Lie Superalgebra $\mathfrak{o}\mathfrak{s}\mathfrak{p}(3\mathrm{\mid }4)$

We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $\mathcal{O}$ for the orthosymplectic Lie superalgebras $\mathfrak{o}\mathfrak{s}\mathfrak{p}(3\mathrm{\mid }4)$ by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.