Congruences modulo powers of 2 and 3 for overpartition k-tuples

Let p̄ₖ(n) denote the number of overpartition k-tuples of n. In 2023, Saikia {saikia} conjectured the following congruences: {align*} p̄q(8n+2)& ≡ 0 {4}, p̄q(8n+3)≡ 0 {8}, p̄q(8n+4) ≡ 0 {2},\\ p̄q(8n+5)& ≡ 0 {8}, p̄q(8n+6) ≡ 0 {8}, p̄q(8n+7)≡ 0 {32}, {align*} where n≥0 and q is prime. Recently, Sellers {sellers2024elementary} showed that these congruences hold for all odd integers q (not necessarily prime). In this paper, we show that the above congruences hold for all positive integers q (not necessarily odd). We also prove the following congruences on OPT̄ₖ(n), the number of overpartition k-tuples with odd parts of n: For all i,j≥ 1, n≥ 0, r not a multiple of 2, k not a multiple of 2 or 3, and not a power of 2, nor a multiple of 2 or 3, we have {align*} OPT̄2^i· r(8n+7)& ≡ 0 {2ⁱ⁺⁴}, OPT̄3^i· 2^j· k(3n+2)& ≡ 0 {3ⁱ⁺¹· 2ʲ⁺²}, OPT̄3^i· 2^j· k(3n+1)& ≡ 0 {3ⁱ· 2ʲ⁺¹},\\ OPT̄3^i·(3n+2)& ≡ 0 {3ⁱ⁺¹· 2}, OPT̄3^i·(3n+1)& ≡ 0 {3ⁱ· 2},{align*} where the first congruence was posed as a conjecture by Sarma et al. {saikiasarma} and the latter four were conjectured by Das et al. {DSS}.