Throughout our lives, we interact daily in conversations with our friends and family, covering a wide range of topics, known as open-domain dialogue. As we age, these interactions may diminish due to changes in social and personal relationships, leading to loneliness in older adults. Conversational companion robots can alleviate this issue by providing daily social support. Large language models (LLMs) offer flexibility for enabling open-domain dialogue in these robots. However, LLMs are typically trained and evaluated on textual data, while robots introduce additional complexity through multi-modal interactions, which has not been explored in prior studies. Moreover, it is crucial to involve older adults in the development of robots to ensure alignment with their needs and expectations. Correspondingly, using iterative participatory design approaches, this paper exposes the challenges of integrating LLMs into conversational robots, deriving from 34 Swedish-speaking older adults' (one-to-one) interactions with a personalized companion robot, built on Furhat robot with GPT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document}3.5. These challenges encompass disruptions in conversations, including frequent interruptions, slow, repetitive, superficial, incoherent, and disengaging responses, language barriers, hallucinations, and outdated information, leading to frustration, confusion, and worry among older adults. Drawing on insights from these challenges, we offer recommendations to enhance the integration of LLMs into conversational robots, encompassing both general suggestions and those tailored to companion robots for older adults.