Quote from Johnny Peppers: "You can restore most of the suppleness to old leather, but you cannot restore the original strength. If it is the original sling I would not depend on it to perform like new leather."
The best thing said so far. You can not expect an old dried out piece of leather to regain it's like new condition.
As far as what to use, avoid at all costs anything with petroleum products or silicone. While some Pecards products have silicone and others do not, ALL contain petroleum and should be avoided on antique leather products that you wish to preserve for posterity, this includes their so called "Antique Leather Care". This holds true for almost all modern "convenient" leather preservatives. If you are going to grease up a pair of work boots, Pecards and so-called mink oils are fine, but not for old leathers that you want to preserve.
The best thing for archival leather items that you want to preserve in original condition is - nothing. Wipe down with a soft, dry cloth to remove loose dust and dirt and leave it as found. Store it in an area where the humidity is a constant 60% and the temperature is strictly regulated. Also avoid direct and prolonged exposure to sunlight.
If you find that you need to use something to restore the oils that have been lost, use a light coating of PURE neatsfoot oil. Yes, it will darken the leather but not too much and certainly no worse than when neatsfoot oil was used on it the last time in military service. And be careful, at all costs avoid the cheap, commonly available neatsfoot oil compound, as I said above, use only pure neatsfoot oil. The compounds are cut with petroleum and mineral oil and will damage leather.