With whom or what are Mrs. Hale and Mrs. Peters in conflict in "A Jury of Her Peers"?

Mrs. Peters and Mrs. Hale are in conflict with the men because of their condescending attitudes.

Shortly after Mr. Hale directs the sheriff and county attorney into the Wrights' home where he has previously discovered that Mr. Wright has been killed, the county attorney has Mr. Hale repeat what occurred the previous day when he and his son came to the Wright's house. Then, he moves toward the stair door, suggesting that he and the other men go upstairs to search for a motive. But, before doing so, he pauses,

"You're convinced there was nothing important here?" he asked the sheriff. "Nothing that would--point to any motive?"

"Nothing here but kitchen things," he [the sheriff] said, with a little laugh for the insignificance of kitchen things. 

Still, the county attorney looks into a high cupboard, and he discovers broken jelly jars that have frozen. "'Here's a nice mess,' he said resentfully." When the women express Mrs. Foster's concern for her preserves, Sheriff Peters laughs,

"Well, can you beat the women! Held for murder, and worrying about her preserves!"

Then, when Mr. Hale laughs and remarks that women are used to worrying about "trifles," the two women move closer to each other. To be polite, the county attorney says something complimentary, but Mrs. Hale and Mrs. Peters do not smile as they feel his words are merely condescending. And so, the men ascend the stairs to the bedroom where Mr. Wright died.

In the meantime, Mrs. Peters gathers up the items that Mrs. Wright has requested be brought to her at the jail. While they look for something to wrap the items, they discover the "trifles in the kitchen" (the broken bird cage and the pretty red box containing a dead canary), trifles that provide the motive for Mrs. Wright's murder of her husband. But, they hide this evidence from the attorney and the sheriff and Mr. Hale, with whom they have been in conflict because they sympathize with Mrs. Wright, whose husband has for years been cold and overbearing and had no respect for her feelings, which were probably "trifling" matters to him. But these were important matters of the heart to Minnie Foster Wright, who once loved to sing and have pretty clothes.