Minimum Sum of Non-Negative Integers with Given Equation - POTW #504

[anemone]

Here is this week's POTW:

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Let $a,\,b,\,c$ and $d$ be non-negative integers.

If $a^2+b^2-cd^2=2022$, find the minimum of $a+b+c+d$.

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[anuttarasammyak]

Thanks for the interesting problem. I have not found the answer but I would like to guess it.
$$s:=a+b+c+d=a+b+d+\frac{a^2+b^2-2022}{d^2}$$
Guessing for minimum s that a=b and $2a^2-2022$ is least with an integer c
$$a=b=32, d=1, c=26$$
$$s=91$$
A nearby case is
$$a=33,b=31,d=1,c=28;\ s=93>91$$
However, for a>>b case
$$a=45,b=1,d=2,c=1;\ s=49$$
My guess failed. I will be glad to know the right answer.