  Unsourced random-access (U-RA) is a type of grant-free random access with a
virtually unlimited number of users, of which only a certain number $K_a$ are
active on the same time slot. Users employ exactly the same codebook, and the
task of the receiver is to decode the list of transmitted messages. Recently a
concatenated coding construction for U-RA on the AWGN channel was presented, in
which a sparse regression code (SPARC) is used as an inner code to create an
effective outer OR-channel. Then an outer code is used to resolve the
multiple-access interference in the OR-MAC. In this work we show that this
concatenated construction can achieve a vanishing per-user error probability in
the limit of large blocklength and a large number of active users at sum-rates
up to the symmetric Shannon capacity, i.e. as long as $K_aR <
0.5\log_2(1+K_a\SNR)$. This extends previous point-to-point optimality results
about SPARCs to the unsourced multiuser scenario. Additionally, we calculate
the algorithmic threshold, that is a bound on the sum-rate up to which the
inner decoding can be done reliably with the low-complexity AMP algorithm.
