FALSE: a conical limit in an enriched category is just a limit in the underlying category

If the statement were true, then conical enriched limits and underlying ordinary limits would coincide.

A concrete failure occurs for $\mathcal{V}=\mathbf{C}\mathbf{a}\mathbf{t}$. Let $\mathcal{M}$ be the strict $2$-category freely generated by objects $a,b$, a $1$-cell $f:a\to b$, and a nonidentity $2$-cell $\alpha :f\Rightarrow f$. Its underlying category is the walking-arrow category, so $b$ is the ordinary product $b\times b$. But a conical product would require $\mathcal{M}(a,b)\cong \mathcal{M}(a,b)\times \mathcal{M}(a,b)$ as categories. Here $\mathcal{M}(a,b)$ is the one-object category with endomorphism monoid $\mathbb{N}$, which is not isomorphic to its square. Thus this ordinary product is not conical.

Hence the converse of [L2] fails: an underlying ordinary limit need not satisfy the enriched conical universal property. The displayed statement is false.