Probabilistic regression with 3-parameters LogNormal distribution ?

Hello everyone,

I'm still learning tensorflow probability and I'm trying to learn by browsing the tutorials and applying them (or trying to) to a real business project.
We usually model our target distribution with a 3-parameter lognormal distribution (or the shifted lognormal) and I'm kind of struggling finding a tweak to the tfd.LogNormal without reimplementing a whole class. For the record, the shifted lognormal is defined this way, with X a real r.v.:
If
then follows a 3-parameters lognormal:

So my idea would be to introduce this shift, as a learnable parameter of the distribution, in the code of the first tutorial, case 2:

model = tf.keras.Sequential([
    tf.keras.layers.Dense(2),  # Maybe a 3rd output ?
    tfp.layers.DistributionLambda(lambda t: tfd.LogNormal(loc=t, scale=1e-3 + tf.math.softplus(0.01 * t[...,1:]))),
    # But then how can I tweak the LogNormal distribution ?
])

I feel that I'm wrong and that is simply not possible.
Any tips here ?

Start with the first tutorial, case 2, and review the tfd.LogNormal and DistributionLambda usage shown in the issue. Determine whether a shifted, learnable three-parameter distribution is an appropriate library feature or requires a new distribution implementation; done means a decided approach with documented behavior and relevant tutorial or tests.