Potential cardinality of a specific minimally unbounded theory

Determine the potential cardinality of the class of countable models (i.e., the size of the set of potential canonical Scott sentences) for T(P,≤,δ) when P consists of an ω‑chain {p_n : n∈ω} with, above each p_n, an antichain {q_{n,m} : m∈ω}, and δ is identically three. Specifically, prove in ZFC that this potential cardinality equals 2^{ℵ0}, as conjectured, or otherwise establish its exact value.

Background

The authors present a minimally unbounded example: P is the union of an ω‑chain {p_n} and, above each p_n, an infinite antichain {q_{n,m}}, with δ identically 3. Under the existence of κ(ω), Theorem 9.11 shows that T(P,≤,δ) is not Borel complete.

To calibrate complexity more finely, the paper employs the notion of potential canonical Scott sentences and their cardinality (the "potential cardinality"). For this example, the authors conjecture that the potential cardinality is the continuum and provable in ZFC, but they lack even a basic upper bound ruling out a proper class.