  Let $D$ be a Fano manifold that may be realised as $\mathbb{P}(\mathcal{E})$
for some rank $2$ holomorphic vector bundle $\mathcal{E}\longrightarrow Z$ over
some Fano manifold $Z$. Let $k\in\mathbb{N}$ divide $c_{1}(D)$. We classify
those K\"ahler cones of dimension $\leq4$ of the form
$(\frac{1}{k}K_{D})^{\times}$ that are smoothable. As a consequence, we find
that any irregular Calabi-Yau cone of dimension $\leq 4$ of this form does not
admit a smoothing, leaving $K_{\mathbb{P}^{2}_{(2)}}^{\times}$ as currently the
only known example of a smoothable irregular Calabi-Yau cone in these
dimensions.
