Let f be a positive definite integral quinary quadratic form. We say f is strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers by f. In this talk, we show that there exist exactly 19 strongly s-regular diagonal quinary quadratic forms representing 1. In particular, we prove the strongly s-regularity of the quinary quadratic form x² + 2y² + 2z² + 3w² + 3t², which is, in fact, of class number 4.