What are the implications of a zero of zeta off the critical line

1. First, let us get history right: Hardy (1914) proved there are infinitely many zeros on the critical line, Hardy-Littlewood (1921) proved there are $\gg T$ zeros on the critical line up to height $T$, and Selberg (1942) proved there are $\gg T\log T$ zeros on the critical line up to height $T$ (i.e. positive proportion).

2. I am quite certain that it is not known that if there exists a nontrivial zero off the critical line, then there are infinitely many such zeros.

Perhaps this is a duplicate of this question If the Riemann Hypothesis fails, must it fail infinitely often?.

The discussion continues here The Hardy Z-function and failure of the Riemann hypothesis, where a hypothesis is proposed, which implies If the Riemann Hypothesis fails, must it fail infinitely often?.

To this hypothesis there are two approximations based on: 1. Zeta function universality 2. GUE hypothesis.

Concerning 1. I can say that even, $\zeta (s)$ for $Re(s)=\frac{1}{2}$ is dense in $\mathbb{C}$, is unknown.

2. it seems to me it will be easier. It is enough to make an inversion for the expression $N(t_{n})=n$, where $N(t_{n})$ the number of non-trivial zeros of the zeta function, which is well-known function and $t_{n}$ the nth non-trivial zero, and define the property of the recurrence.