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Keep the two quantile identities
The original loss record has q95=16.4485362695... and q99=23.2634787404..., in one common loss unit and horizon. Corresponding standard normal z values are 1.64485362695... and 2.32634787404....
Subtract before dividing
The quantile difference is 10 times the standard z difference. Dividing those differences gives σL=10. Use unrounded values where available rather than treating displayed six-decimal figures as exact evidence.
Recover the mean
Mean loss is q95−10×z95=0. Substituting μL=0 and σL=10 into both confidence equations reproduces the supplied quantiles. A negative inferred scale would contradict the stated positive-scale setup.
Do not call the reconstruction a normality test
Two numerical summaries can fit two normal-model parameters without proving the whole distribution is normal. This is an original algebraic consistency task, not an estimator from observed market outcomes.