I found out this conjecture which is tested for $m\leq 100$:
For all natural numbers $n>\!30$, for all $\alpha\in\mathbb Z$ and all $\mu=0,\pm 1,\pm 2$:
$m\equiv p_n^2+\mu p_n+\alpha\!\pmod{p_{n+1}}$ $\wedge\; m<p_{n+1}\iff$ $\exists k\in\mathbb Z_+:m=4k^2-2\mu k +\alpha$
Since this is a more general case than in the question For prime numbers $p_n$ it holds $m\equiv p_n^2\pmod {p_{n+1}}\implies$ $m$ is a square and that (due to an answer) was related to an unproved conjecture about prime gaps, I suspect that this conjecture is also related to prime gaps and is to deep to be proved.
I would like to know:
Can this conjecture relate to prime gaps?
Are there counterexamples?
Do someone have an intuition about the logic structure of the conjecture?
Some context:
I use bigz implemented in ANS-Forth with numbers and sets on stacks and all operations post-fix. First the case with $\alpha=\mu=0$:
1 100 :| n | n pnr@ ^2 n 1+ pnr@ umod dup sqr and ; build-set cr z.
{0,1,4,16,36,64,100,144,196,324} ok
DUP SQR AND mask the result from UMOD so all non square results are transformed to 0. SQR results in $-1$ or $0$ depending on if the top of the number stack is a perfect square or not. Z. prints (and consumes) the set of the top of the set stack.
30 1000 :| n | n pnr@ ^2 n 1+ pnr@ umod dup sqr and ; build-set cr z.
{0,4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024,1156} ok
Still exceptions indicated by 0 when $n\geq 30$.
31 1000 :| n | n pnr@ ^2 n 1+ pnr@ umod dup sqr and ; build-set cr z.
{4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024,1156} ok
No exceptions when $n>31$.
31 10000 :| n | n pnr@ ^2 n 1+ pnr@ umod dup sqr and ; build-set cr z.
{4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024,1156,1296,1444,1600,1764,1936,2116,2304,2500,2704,2916,3136,3364,3600,3844,4096,5184} ok
31 50000 :| n | n pnr@ ^2 n 1+ pnr@ umod dup sqr and ; build-set cr z.
{4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024,1156,1296,1444,1600,1764,1936,2116,2304,2500,2704,2916,3136,3364,3600,3844,4096,4356,4624,4900,5184,5476,5776,6084,6400,6724,7056,7396,7744,8100,9216,9604,10000,12544,12996} ok
The unbroken sequence of $4k^2$ values get longer for greater intervalls.
Now an example with $\alpha=3$ and $\mu=1$:
31 10000 :| n | n pnr@ ^2 n pnr@ + 3 + n 1+ pnr@ umod ; build-set cr z.
{5,15,33,59,93,135,185,243,309,383,465,555,653,759,873,995,1125,1263,1409,1563,1725,1895,2073,2259,2453,2655,2865,3083,3309,3543,3785,4035,5115} ok
And an example with $\alpha=7$ and $\mu=-2$:
31 10000 :| n | n pnr@ ^2 n pnr@ 2* - 7 + n 1+ pnr@ umod ; build-set cr z.
{15,31,55,87,127,175,231,295,367,447,535,631,735,847,967,1095,1231,1375,1527,1687,1855,2031,2215,2407,2607,2815,3031,3255,3487,3727,3975,4231,5335} ok