The divergence conjecture for Fibonacci and arithmetic-progression sequences

Let $F$ denote the Fibonacci numbers, let $AP_F$ denote the associated arithmetic-progression sequence, and let $\Delta(L,k;r)$ and $n(AP_F,k;r)$ be the parameters defined in the paper for the corresponding Ramsey-type problems. The notation $\Delta(F,3;4)$ denotes the first parameter for Fibonacci sequences with the indicated values, while $n(AP_F,4;2)$ denotes the corresponding parameter for $AP_F$. Divergence conjecture.

$$\Delta(F,3;4)=n(AP_F,4;2)=\infty.$$

The conjecture is motivated by the large gaps between the known values and bounds for nearby parameters: the paper proves $\Delta(F,4;4)=\infty$ and $n(AP_F,5;2)=\infty$, while computational results give $\Delta(F,3;2)>100000$ and $n(AP_F,4;2)>8000$.