Sharpening of Szabó's conjecture for arithmetic-progression-intersection families

Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t\subseteq\\{1,\dots,N\\}$ such that $A_i\cap A_j$ is a nonempty arithmetic progression for all $i\ne j$. A construction gives

$$t(N)\ge \binom{N}{2}+1+\left\lfloor\frac{N-1}{4}\right\rfloor.$$

Sharpening of Szabó's conjecture. Equality holds in this lower bound for every $N\ge 1$, that is,

$$t(N)=\binom{N}{2}+1+\left\lfloor\frac{N-1}{4}\right\rfloor.$$

The claim is verified computationally for $N\le 12$ and would imply the strongest form of Szabó's bound, while the remaining obstruction is the kernel question concerning whether an extremal family has a common element.