The basic idea is to rewrite the swaption payoff as a option on a sum of bonds to a sum of options.

To recognize the bond dependence on HW model we rewrite Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator.

By analogy

Recall that  Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator. is "model parameter". We can find x* which makes swap value zero.

When this value is zero, we can add it for free to main swaption:

By rewriting each swap in legs we finish by:

Now, the most important step is:

Look at the swap value, recall that Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator. is fixed, and other components which depend on x become smaller when Unknown macro: formula. Click on this message for details.
The [formula] macro is not in the list of registered macros. Verify the spelling or contact your administrator. thus the total swap value becomes greater. This allow us to be sure that we can replace optionality by indicator and then open the brackets.

What if some coefficient has opposite sign?

[In progress..] Jamshidian's trick is still valid if the coefficients change the sign once.