Below is the answer from a leading Statistician in our industry.
You are exactly correct. As you can see in the output, the Minimum Significant Difference at 90% confidence is 0.3831 and the Minimum Significant Difference at 95% confidence is 0.4382. So the sample means need to be farther apart to conclude that they are significantly different at the 95% confidence level than they need to be at the 90% confidence level.
However, your client might be asking a slightly different question. When you round the samples means to a reasonable number of decimal places (1), you get: