Spira's Riemann-hypothesis conjecture for sections of the Hardy Z-function

Let $Z_{N(t)}(t)$ denote the section of the approximation to the Hardy $Z$-function used in Spira's formula, where

$$N(t)=\left[\frac{t}{2}\right].$$

A zero is called non-trivial in the usual sense of the zeros associated with the zeta function, and a zero of $Z_{N(t)}(t)$ is real when its argument $t$ is real.

Spira's conjecture. All the non-trivial zeros of $Z_{N(t)}(t)$ are real.

This is presented as a formalization of Spira's empirical observation that the higher-range approximation appears not to admit zeros off the real line, suggesting a connection with the Riemann hypothesis. The source gives no resolution of the conjecture.