Parallel LTC amplification conjecture

Let $X$ be a family of high-dimensional expanders and let the associated locally testable codes admit a $q$-step random walk or distribution generating $k$ valid parallel tests. The resulting parallel tester is obtained by running these $k$ tests in parallel.

Parallel LTC amplification conjecture. There exists such a family for which the construction gives a parallel variant of the theorem on LTC amplification:

1. There is a $q$-step random walk or distribution on $X$ generating $k$ valid parallel tests.
2. The resulting parallel tester has no asymptotic decay in soundness.

The conjecture asks whether high-dimensional expanders and locally testable codes can support parallel simulation without the soundness deteriorating asymptotically. It is motivated by agreement tests on dense complexes such as Grassmannians and spherical buildings, but the statement is presented as an informal conjecture and its resolution is not supplied here.