It's a bit of a head-scratcher, but I had a go at playing "devil's advocate" and wrote to both the show host and the PhD student advocating a new form of social Trinitarianism to try and assess the strengths of this approach.
Here then I posit the following impossible triad (all cannot be true) and how I think a Trinitarian should answer. These alternatives are inspired by Jurgen Moltmann's distinction that it is not as accurate to say death of God as death in God (The Crucified God) and McIntosh's intrinsic/group persons. If I were a fourth century or later trinitarian, I would also want to distinguish between person and being, or intrinsic and group persons. I would say that the Triune God is a being (or group person) and that Jesus Christ is not a being (or a group person); Jesus Christ is an intrinsic person.
God = one (group) being; God = three fully divine intrinsic persons, F S & HS
Immortal = "never dying"
1) God is essentially immortal
2) No fully divine person has ever died
2) Jesus is fully divine
As I mentioned in my comment, I think the way forward for a capital T Trinitarian might first be to deny the wording as accurate because Jesus Christ is an intrinsic person, not a being (i.e. human-divine person, not a human being), to substitute the word person for being, then deny 2. Now they can take refuge in the person/being distinction and propulse a possible further distinction that might follow from Moltmann's thought, that God experienced death within him.
Alternatively, if we took a McIntosh Group Person social Trinity, this scenario could invite the comparison with a closely knit family losing a treasured member. The functional group person experiences the death of an intrinsic person. Here, it is the group person who is essentially immortal, and the intrinsic person Logos incarnandus (Barth) who is not.