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ACROSS

 1 Alert: Spanish uncle vanished with slip of the lip over my dead body
(3,2,1,3,4,2)

 9 Fed reorganized, amid loud call for strange political ally? (9)

10 Mathematician’s error breaking rule (5)

11 Circle swimming hole the other way (4)

12 Uniformed students overdose on drinks, backing up propagandist
(4,6)

14 Heard (perceived) part of show (5)

15 Engineer dictates conclusive assessment (4,4)

18 Exceedingly shoddy mimic in box (8)

20 Third of December: frightful date for military trainee (5)

23 Diner with lounge atmosphere, not excluding oxygen (10)

25 Formal park boundary (4)

27 A large quantity of boys grabbing ring (5)

28 50 imprisoned in chains with incomplete plan to deliver a nasty surprise
(9)

29 Compilation of letters from maligned YA author (9,6)

DOWN

 1 Singer, for example, with a number on live record (8)

 2 Arrange to plead for slippery youngster? (7)

 3 Oral disease moved quickly (4)

 4 I teach pal mysterious Muslim rule (9)

 5 Aware, naked, and hopeless (2-3)

 6 Symbol pinned to your ear is a critical feature of many conversations
(3,7)

 7 Exercise regimen for Pontius and his family? (7)

 8 Spanish performer, ripped or flipped (6)

13 Bone and grain maintaining Turkic population (10)

16 Highly concentrated incentive discussed after trade (9)

17 Mad, steep chaos! (8)

19 Carefully conserve a set of cards securing computer connection (7)

21 Head of livestock entrusted to brave pet (7)

22 Phase originally boosting runner! (6)

24 Barbie nearly failing as congregation’s leader (5)

26 British comic is an object of worship in audio recording (4)

SOLUTION TO PUZZLE NO. 3499

ACROSS 1 & 4 LE + GAL + DOUB(LE)TS
9 ME(GAP)H + ONE
11 NU[tshe]LL {+VOID}
12 PER + SU + ASIV-E (rev.)
15 [t/L]-OI-[l/T]-ER 16 SOS[o]
18 GA(LO)SH 20 HE(IF)ER (here anag.)
23 POST(P + ON)ING 24 hidden
26 C(E)ASE {+DESIST} 27 HO(I + POLL)OI (Ohio anag.) 29 DE-[f/BU]-G

DOWN 1 LEMON(A)DE 2 GIG OLOS (rev.) 3 LAP + D (rev.) 4 DO(O)VE + R 5 U(TENSIL[e])S 6 anag. 7 T(HEW)IRE 8 F(RE)E {+CLEAR}
14 SHOW(PIE)CE (chew so anag.) 16 SHANG(H)AI (Ghana is anag.)
17 B + RE(AKIN)G {+ENTERING} 19 [c]AB + STAIN 21 F(ANC)LUB
22 [l]ENDING 23 2 defs.