Is there a way to model data that are skew normally distributed, but for which one builds in two seperate standard deviations?

The parameter σ_1 should specify the 15.9% to 50% interval, whereas σ_2 should specify the 50% to 84.1% interval (i.e., the middle 68.2% of values).

The idea is that the values of σ_1 and σ_2 should be computed from the data; and together with the mean, give parameters to plot the representative probability density function. The result will look skew normally distributed, unless σ_1 = σ_2, in which case the PDF would be modeled as a normal curve. Importantly, the area under the probability density curve between σ_1 and the mean as well as the area between the mean and σ_2 should both be 34.1% of the probability and 68.2% when combined.

Note that the skew normal distribution, the log-normal distribution and the Raleigh distribution does not seem to allow this trivially as they don't have two such σ_1 and σ_2 parameters.

An example for which σ_2 > σ_1: