In a story of this type there does not have to be a conflict between characters or a character's inner conflict. This is a story about the greatest conflict in the history of mankind, and it might be viewed as a sort of coda to an epic tale about the atomic holocaust. The war has nearly destroyed everything. As far as we know, these four men may be the only people left alive on the entire planet. If so, then humanity is obviously doomed, since there are no surviving women to produce babies--if any woman would want to bring a child into such a world. The four men have enough of a conflict just trying to stay alive in this hostile environment. In the end it seems as if they are likely to turn against one another in spite of their apparent effort to preserve some vestiges of civilization. One of the guests who enjoyed the literary and musical soiree appears to be loitering nearby outside, waiting for the host to fall asleep so that he can creep in and steal some or all of his treasures--the books, the record albums, and the portable phonograph.
At last he prayed, and got in under his blankets, and closed his smoke-smarting eyes. On the inside of the bed, next the wall, he could feel with his hand the comfortable piece of lead pipe.
So there is a minor conflict of man against nature, which the men are sure to lose, and of man against man, which is inevitable.
The fundamental conflict in the short story goes to art and one's appreciation of it. I think that there is an evident conflict of survival in a post-apocalyptic world. Yet, I think that the relationship between Jenkins and the musician might be the basis for the strongest conflict present. In a world where art has been destroyed and in a setting where art is no longer easily appreciated, Jenkins "has" a chance at "more soul" than the others because he possesses art. The phonograph as well as his books symbolizes an ability to appreciate art, something that is otherwise impossible to accomplish in the setting of the short story. When the musician listens to the Debussy nocturne, he has to leave because this is an artistic expression and requires an ability to appreciate it that he lacks. It is this conflict that resides between the haves (Jenkins) and the have nots (the musician.) The ending where Jenkins sleeps with the metal pipe in one hand represents this conflict of power even in a post- apocalyptic setting.