Boolean symmetric vs. functional PCSP dichotomy

Given a 3-uniform hypergraph $(V,E)$ that is promised to admit a $\{0,1\}$-colouring such that every edge contains exactly one $1$, can one find a $d$-colouring $h:V\to \{0,1,\ldots,d-1\}$ such that $h(e)\in R$ for every $e\in E$? This can be cast as a promise constraint satisfaction problem (PCSP) of the form $\operatorname{PCSP}(1-in-3,\mathbf{B})$, where $\mathbf{B}$ defines the relation $R$, and is an example of $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ (and thus wlog also $\mathbf{B}$) is symmetric. The computational complexity of such problems is understood for $\mathbf{A}$ and $\mathbf{B}$ on Boolean domains by the work of Ficak, Kozik, Ol\v{s}\'{a}k, and Stankiewicz [ICALP'19]. As our first result, we establish a dichotomy for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ is Boolean and symmetric and $\mathbf{B}$ is functional (on a domain of any size); i.e, all but one element of any tuple in a relation in $\mathbf{B}$ determine the last element. This includes PCSPs of the form $\operatorname{PCSP}(q-in-r,\mathbf{B})$, where $\mathbf{B}$ is functional, thus making progress towards a classification of $\operatorname{PCSP}(1-in-3,\mathbf{B})$, which were studied by Barto, Battistelli, and Berg [STACS'21] for $\mathbf{B}$ on three-element domains. As our second result, we show that for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ contains a single Boolean symmetric relation and $\mathbf{B}$ is arbitrary (and thus not necessarily functional), the combined basic linear programmin relaxation (BLP) and the affine integer programming relaxation (AIP) of Brakensiek et al. [SICOMP'20] is no more powerful than the (in general strictly weaker) AIP relaxation of Brakensiek and Guruswami [SICOMP'21]