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The pdf $f_V$ is a mixture of the family $\F_A:=(f_a)_{a\in[-A,A]}$ of the normal pdfs $f_a$ with mean $a$ and variance $1$. By Theorem 6 on p. 2129 of Kemperman, for each natural $s$, the pdf $f_V$ will have at most $s$ modes (or, more precisely, modal intervals) iff any mixture of any $2s$ members of the family $\F_A$ has at most $s$ modal intervals.

This theorem is a generalization of Theorem 4 on p. 2128 of Kemperman, which implies that all mixtures of a family of pdfs are unimodal iff any mixture of any two members of the family is unimodal.

Further, by Remark 1 on p. 2133 of Kemperman, our pdf $f_V$ will be unimodal for all admissible $U$ iff for all $a,b,x$ such that $-A\le a<x<b\le A$ we have
\begin{equation}
f'_a(x)f''_b(x)\ge f'_b(x)f''_a(x),
\end{equation}
which can be rewritten as $(a-x)(b-x)\ge-1$ for all such $a,b,x$, which is equivalent to $A\le1$.

The case of more modes than $1$ appears much more difficult analytically.