It will be up to the comittee to decide if holes 6 and 10 share the same green, or if they have two seperate, but adjoining greens.
If they share the same green:
Hole 10 (if the player is playing hole 6) is a “Hole made by a greenskeeper” and is ground under repair (dec 16/7). Rule 25-1-b-iii states that if the ball is in the hole, or the hole is in the line of putt, then the player is entitled to free relief as follows.
(iii) On the Putting Green: If the ball lies on the putting green, the player must lift the ball and place it without penalty at the nearest point of relief that is not in a hazard, or if complete relief is impossible, at the nearest position to where it lay that affords maximum available relief from the condition, but not nearer the hole and not in a hazard. The nearest point of relief or maximum available relief may be off the putting green.
If Hole 10 is on a different green, then the ball is on a “wrong putting green” 25-3-b decrees the player must proceed as follows:
If a player’s ball lies on a wrong putting green he must not play the ball as it lies. He must take relief, without penalty, as follows:
The player must lift the ball and drop it within one club-length of and not nearer the hole than the nearest point of relief. The nearest point of relief must not be in a hazard or on a putting green. When dropping the ball within one club-length of the nearest point of relief, the ball must first strike a part of the course at a spot that avoids interference by the wrong putting green and is not in a hazard and not on a putting green. The ball may be cleaned when lifted under this Rule.
(Note, these are USGA rules. R&A rules apply in the “British” Open Championship, but they should be substantially similar)