Monotonic motion conjecture for zeros of interpolated Xi-approximates

Let $\Xi_k(z)=\sum_{j=1}^{k}\Phi_j(z)$ and let $\Phi_{k+1}(z)$ be the next summand in the truncated Riemann Xi-function series. For $k\geq1$ and $0\leq t\leq1$, consider $\Xi_k(z)+t\Phi_{k+1}(z)$. Monotonic motion conjecture. The imaginary part of each non-real zero of $\Xi_k(z)+t\Phi_{k+1}(z)$ decreases monotonically, continuously, as $t$ goes from $0$ to $1$. The claim is based on computational indications in the source; no proof or resolution is supplied.