Deed of Variation, gift of mortgaged property, & SDLT

I have looked at the previous posts on similar topics and just want to check my understanding please.

A has died leaving whole estate including house to B. House is subject to equity release mortgage.

B wishes to execute a Deed of Variation making a gift of the house, subject to the mortgage, to C.

C intends to take out his own mortgage on the house in order to repay the equity release mortgage.

My understanding is that C incurs no SDLT on this transaction, as the assumption of a pre-existing debt does not count as external consideration for this purpose (Finance Act 2003 Sch 3 3A(4). Do the forum members agree or am I missing something obvious!

Steve Carter