Representations of the super-Yangian of type $B(n,m)$

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${\mathfrak{o}sp}_{2n+1\mathrm{\mid }2m}$. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case $n=1$ and arbitrary $m⩾1$.