This is false for all $a \leq \frac{1}{4}$. To see this, glue two copies of $K_{\ell, \ell}$ together at a vertex $x$. This is a bipartite graph with $4\ell-1$ vertices and minimum degree $\ell$. However, the vertex-connectivity is $1$ since $x$ is a cut-vertex.
Tony Huynh
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