When developing loads for my Pedersoli Kodiak Safari .72-caliber double rifles, I chose a custom conical bullet as its payload. Since I was loading heavily by the standard of Pedersoli's attorneys, I wanted to do two related things up front: proof each barrel; be certain that the UNFIRED conical would not be jarred into a dangerous position by severe recoil from fired barrel.
Proofing went as expected. To verify conicals would not move once loaded, I loaded both barrels and marked my loading rod for each barrel's actual seating depth (Depths are within hundredths of an inch of each other). I fired 20 rounds from barrel #1 only, using my "dip stick loading rod to check barrel #2's bullet position. After 20 rounds, I shot barrel #2 without issue. Process was reversed. . . . This required two shooting sessions - heavy loads!
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Okay, to the punch lines. Rather than guess about conical bullets' moving, verify it. The bullet I use in all of my Big Bores is intended to be mallet-loaded. In the instance of a double rifle, a repeat shot is not an issue. Shot #3 will be delayed about five minutes.
If your barrel has choke or taper bore, you may be able to load a normally mallet-loaded conical with a firm pop on a short starter. But again, verify.
Hope this helps.