Using the Switching regression and Instrumental Variables (IV) in the same model

Dear Statalist,

I have been looking at the switching regression to identify public-private wage differences:

for public sector workers: log(public_wage) = A + B*X1 + E1
for private sector workers: log(private_wage) = C + D*X2 + E2

selection equation is: I = G((log(public_wage) - log(private_wage)) + H*Z1 * J*Z2 + U

where B, D, G, H and J are parameters to be estimated, E1, E2 and U are error terms, A and C are constant terms. ZI and Z2 are exclusion restrictions such as marital status and spouse's sector of employment that are included in the selection equation but do not directly affect wages (the first 2 equations).
I is a latent variable that determines the sector an individual is employed in. If I>0, the individual works in the public sector, otherwise the private sector.

Although this helps me tackle the issue of sample selection, I wish to test for endogeneity of the exclusion restrictions similar to what IV does. I have found in the literature that this has been done using the Heckman probit model and IV, but have not come across anything for the switching regression.

This was suggested to me:
Instead of starting with a switching regression, have an endogenous dummy variable model - the first stage runs wage regressions for public and private workers separately. The second stage is where the public-private sector is measured as a dummy (1=private, or 0 otherwise) allowing me to test for the strong instruments (Z1 and Z2). Ideally, I want to apply the endogenous dummy variable literature to the switching regression.

I am not sure how this can be done and would appreciate any advice on this.

Thank you