Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ the subspace of those tensors $s$ such that $\int_M \operatorname{tr}_g(s) \, \mathrm{d} V = 0$ and finally, let $E$ be the subspace of $S^2(M)$ formed by those $s$ such that $s_x(N(x), v) = 0$ for all $x \in \partial M$ and $v \in T_x(\partial M)$.

If $h \in S^2_0(M)$, then there exists a smooth $1$-parameter family of metrics $g_t$ such that

1. $g_0 = g$;
2. $g_t$ has constant volume; and
3. $\frac{\mathrm{d}}{\mathrm{d}t} \big\vert_{t=0} g_t = h$.

It suffices to define $$g_t = \frac{g + th}{V(g + th)^{2/n}},$$ where $V(\cdot)$ denotes the volume.

My question is: given $h \in S^2_0(M) \cap E$, does there exist a smooth $1$-parameter family of metrics $g_t$ such that

1. $g_0 = g$;
2. $g_t$ has constant volume;
3. $\frac{\mathrm{d}}{\mathrm{d}t} \big\vert_{t=0} g_t = h$; and
4. if $h_t := \frac{\mathrm{d}}{\mathrm{d}t} g_t$ and $N_t$ is the outward unit normal to $\partial M$ with respect to the metric $g_t$, then $h_t(N_t, v) = 0$ for all $v$ tangent to $\partial M$ and for all $t$

?