Lognormal family in brms with true identity link

So it seems that the “identity” link in brmslognomal family isn’t actually an identity link (that is, E[y|X]\neq bX, but instead E[log(y)|X] = bX:

library(brms)

mlognorm <- brm(mpg ~ hp + wt, 
                family = lognormal(link = "identity"), 
                data = mtcars)

mnorm_log <- brm(log(mpg) ~ hp + wt, 
                 family = gaussian(), 
                 data = mtcars)

parameters::model_parameters(mlognorm, test = NULL)
#> Parameter   |    Median |         95% CI |  Rhat |     ESS
#> ----------------------------------------------------------
#> (Intercept) |      3.83 | [ 3.68,  3.96] | 1.001 | 1385.00
#> hp          | -1.52e-03 | [ 0.00,  0.00] | 1.000 | 1458.00
#> wt          |     -0.20 | [-0.26, -0.15] | 1.001 | 1167.00

parameters::model_parameters(mnorm_log, test = NULL)
#> Parameter   |    Median |         95% CI |  Rhat |     ESS
#> ----------------------------------------------------------
#> (Intercept) |      3.83 | [ 3.68,  3.96] | 1.001 | 1379.00
#> hp          | -1.57e-03 | [ 0.00,  0.00] | 0.999 | 1684.00
#> wt          |     -0.20 | [-0.26, -0.15] | 1.000 | 1199.00

I would say that the calling this an identity-link is (1) misleading, and (2) this means we aren’t modeling E[Y] but E[log(Y)], which I am uneasy about.

Is there a way to have actually model a linear relationship between X and Y with Y being conditionally lognormally distributed using brms?

Okay, solved here: