Your conjecture is correct, but the answer is going to be somewhat technical.

First of all, we are going to make the Ansatz that the solution is actually smoother wrt time (this will be justified a posteriori), so that the boundary condition can be differentiated wrt to time to yield the dynamic b.c.
$$\frac{\partial Lu}{\partial t}(t,x)=a(t,x)$$
where the boundary operator $L$ is defined by
$$Lu(t,x):=\frac{\partial u}{\partial \nu}(t,x)+b(t,x)u(t,x).$$
In this way, you can turn your problem into an abstract Cauchy problem
$$
\frac{d\mathcal u}{d t}(t)={\mathcal A}{\mathcal u}(t)+\Phi(t)
$$
where $\mathcal A$ is the operator matrix
$$
\mathcal A:=\begin{pmatrix}\Delta & 0\\ 0 & 0\end{pmatrix}
$$
with domain
$$
D(\mathcal A):=\left\{\begin{pmatrix}u\\ Lu\end{pmatrix}:u\in H^\frac{3}{2}(\Omega):\Delta u\in L^2(\Omega) \right\}
$$
and the inhomogeneous term is
$$
\Phi(t):=\begin{pmatrix}0\\ a(t,\cdot)\end{pmatrix}
$$
Now, you are in the setting of the Klaus-Jochen Engel's theory of *one-sided coupled operator matrices* and you can apply a result - essentially due to Engel, but you can find it in a ready-to-use-version in an old paper by Kramar and myself (Proposition 8.10 in "Theory and applications of one-sided coupled operator matrices", 2003) that claims that positivity (and in fact irreducibility) of the semigroup $e^{t\mathcal A}$ generated by $\mathcal A$ follows from positivity (and in fact irreducibility) of the semigroup generated by $\Delta$ with homogeneous Robin boundary conditions plus the strong maximum principle that holds for the Helmholtz equation with homogeneous Robin boundary conditions. (If you are not very familiar with the theory of operator semigroups: positivity and irreducibility of the semigroup generated by $\mathcal A$ correspond to an affirmative answer to your question in the case $a\equiv 0$.)
(Everything so far is precisely true in the one-dimensional case: in the $n$-dimensional case you need a slight generalization of Engel's theory presented in a more recent paper by myself, "Asymptotics of semigroups generated by operator matrices", 2014.)

Now, validity of your conjecture simply follows from the Duhamel formula that tells you that
$$
{\mathcal u}(t)=e^{t\mathcal A}{\mathcal u}_0+\int_0^t e^{(t-s)\mathcal A}\Phi(s)\ ds\ .
$$